We present direct methods and symbolic software for the computation of conservation laws of nonlinear partial differential equations (PDEs) and differential-difference equations (DDEs).The methods are applied to nonlinear PDEs in (1+1) dimensions wit
h polynomial nonlinearities which include the Korteweg-de Vries (KdV), Boussinesq, and Drinfeld-Sokolov-Wilson equations. An adaptation of the methods is applied to PDEs with transcendental nonlinearities. Examples include the sine-Gordon, sinh-Gordon, and Liouville equations. With respect to nonlinear DDEs, our methods are applied to Kac-van Moerbeke, Toda, and Ablowitz-Ladik lattices. To overcome the shortcomings of the undetermined coefficients method, we designed a new direct method which uses leading order analysis. That method is applied to discretizations of the KdV and modified KdV equations, and a combination thereof. Additional examples include lattices due to Bogoyavlenskii, Belov-Chaltikian, and Blaszak-Marciniak. The undetermined coefficient methods for PDEs and DDEs have been implemented in Mathematica. The code TransPDEDensityFlux.m computes densities and fluxes of systems of PDEs with or without transcendental nonlinearities. The code DDEDensityFlux.m does the same for polynomial nonlinear DDEs. Starting from the leading order terms, the new Maple library discrete computes densities and fluxes of nonlinear DDEs.
